Tensorial and Hadamard product inequalities for functions of selfadjoint operators in Hilbert spaces in terms of Kantorovich ratio

Abstract

Let H be a Hilbert space. In this paper we show among others that, if f, g are continuous on the interval I with   0 <γ ≤ f(t)/g(t)≤ Γ for t ∈ I   and if A and B are selfadjoint operators with Sp (A), Sp (B) ⊂ I, then     [f1−ν(A) gν (A)] ⊗ [fν(B) g1−ν (B)] ≤ (1 − ν) f (A) ⊗ g (B) + ν g (A) ⊗ f (B)                 ≤ [(γ + Γ) 2/4γΓ]R [f1−ν(A) gν (A)] ⊗ [fν(B) g1−ν (B)].   The above inequalities also hold for the Hadamard product “ ◦ ” instead of tensorial product “ ⊗ ”.